Integral Geometry and Tomography

The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they are quite useful for dealing with inverse problems in hyperbolic differential equations (integrals of unknown coefficients over ellipsoids or lines can be obtained from the first terms of the asymptotic expansion of rapidly oscillating solutions and information about first arrival times of a wave). While there has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is unity, the situation is not quite clear even when the weight function is monotone along, say, straight lines in the plane case (attenuation). We give a brief review of this area, referring for more information to the book of Natterer [Nat].